Nematic order parameter

The first ingredient of any theoretical description (and simulation) is some notion of an order parameter to distinguish between the isotropic and nematic phases.

In Fig. 1.1(b) we introduced the director ~n, giving in which direction molecules preferentially align. However, we have not defined a measure of how strongly aligned they are in that direction, nor have we shown how to calculate ~n given a set of molecular orientations. To this end, consider the arrangements of particles in the isotropic and nematic phases of Fig. 1.1. One might naively guess that a scalar order parameter suffices to distinguish between the isotropic and nematic phase. However, more complicated phases (such as biaxial nematics) exist and it turns out that a scalar is insufficient to fully describe the nature of nematic ordering. In fact, the nematic order parameter takes the form of atensor.

Definition of the orientational tensor

The form of the tensor follows quite naturally when one tries to compute the nematic director ~n for a system of i = 1, . . . , N “rod-like” molecules. The orientation of molecule i is given by a normalized vector d~_i = (x_i, y_i, z_i) so that |d~_i| = 1. The director~n is given by that vector for which the projection

P(~n) = 1

Figure 1.6: Schematic representation showing the difference in molecular order be-tween (a) the “standard” nematic (S > 0) where particles align parallel to the director ~n and (b) the “perpendicular” nematic (S < 0) where the particles align ⊥to~n.

is maximized. Note the presence of the square in the definition of P(~n) which is required by the inversion symmetry, d~_i ↔ −d~_i, of the nematic phase. Finding the vector ~n that maximizes P yields an Euler-Lagrange problem with constraint

|~n| = 1. This can be cast into an eigenvalue problem, Q·~n = λ~n, with eigenvalue λ, and where the tensorQ is given by

Q= 3 2





hx²i −1/3 hxyi hxzi hxyi hy²i −1/3 hyzi hxzi hyzi hz²i −1/3



, (1.3)

and where the angular brackets denote averages over all molecular orientations hαβi ≡ 1

N

N

X

i=1

α_iβ_i, (α, β) = (x, y, z). (1.4) Note that the trace ofQ is zero, since the molecular orientations are assumed to be normalized.

The ordering of nematic phases is thus encoded in the orientational tensor Q via its eigenvectors and eigenvalues. Note thatQ can alternatively be expressed in its more usual “short form” as

Q_αβ = 3

2hαβi −1

2δ_αβ, (1.5)

with Kronecker-delta δ_αβ. In addition, we point out that the definition of Q in Eq.(1.3) is very convenient from the point of view of simulations since the molecular orientations d~_i are explicitly stored in memory.

1.4 Isotropic-nematic transition: bulk theoretical approaches

Relation between the orientational tensor and nematic structure

To distinguish between isotropic and nematic phases, one needs to computeQ and bring it into diagonal form. SinceQ is symmetric it follows that all eigenvalues are real. Note also that the trace of Q is zero. In the isotropic phase, one trivially obtains a null matrix¹

while a (uniaxial) nematic phase yields

Q^diag_nem =S It is also possible to have a nematic phase where the particles align perpendicularly to the director, see Fig. 1.6. In that case one finds

Q^diag_⊥ =S i.e. the same as for the uniaxial nematic, but with a negative prefactor. Finally, for a biaxial nematic, one finds that

Q^diag_biaxial =

where η measures the degree of biaxiality (when η = 0 one recovers the nematic form ofQ again).

Scalar nematic order parameter

From the above examples it follows that a scalarS is sufficient to describe isotropic (S = 0), “standard” nematic (S >0), and “perpendicular” nematic (S < 0) phases, but not biaxial nematics. Fortunately, biaxial nematics are relatively scarce, and we do not consider them in this thesis. Hence, we will mostly use the scalar S as nematic order parameter in what follows. In simulations, this requires one to compute the eigenvalues of the orientational tensorQ: the signs of the eigenvalues tell us whether the phase is nematic, and the prefactor of Q tells us how strongly the nematic phase is aligned (see additional discussion in [47]).

1Of course, for the isotropic phase, diagonalization ofQis not necessary.

Note that S is related to the angular distribution function f(θ, φ) via

where f(θ, φ) denotes the probability of a molecule pointing in the direction (θ, φ) where 0< θ < π and 0< φ <2π are the “standard angles” of spherical coordinates.

As f(θ, φ) is a probability, we require a normalization condition Z Note that it is implicitly assumed in Eq.(1.10) that the director corresponding to f(θ, φ) is parallel to thez-axes.

Measurement of the nematic order parameter in simulations

In our computer simulations we have direct access to the molecular orientations:

the order parameterS can thus be calculated explicitly from the eigenvalues of the orientational tensor Q. The nematic order parameter follows from the eigenvalues of the orientational tensor Q, which involves the diagonalization of a 3×3 matrix.

The tensorQ is trivially computed in simulations using Eq.(1.5) and the molecular orientations d~_i. The diagonalization is performed using the exact expression for the roots of a cubic polynomial. Comparing to Eq.(1.7), there are several choices to extract S. The usual definition of S is simply to take the largest eigenvalue of Q; the nematic director ~n is given by the corresponding eigenvector. However, in the isotropic phase, this choice leads to a finite-size artifact. If always the largest eigenvalue is taken, one also finds a finite nematic order parameter in the isotropic phase of order O N^−1/2

, with N the total number of molecules [48]. We therefore also occasionally use an alternative definition by taking S to be −2× the middle eigenvalue of Q. As can be seen from Eq.(1.7), this choice also constitutes a valid definition of S, which has the advantage of yielding S = 0 when averaged in the isotropic phase.

Measurement of the nematic order parameter in experiments

In experiments the above method of calculating the nematic order parameter is usu-ally not possible (except in systems of colloidal rods where real-space resolution is available [49]). There are two fundamentally different ways the nematic order pa-rameter can be measured in experiments. One can measure the anisotropy of some macroscopic function, for example magnetically [50], electrically [51], or optically [51], and the only additional knowledge required is of the maximum possible mag-nitude of the function being measured, i.e. the anisotropy of the perfectly ordered nematic phase. Birefringence is a good example of such a measure. The most sim-ple case involves materials with uniaxial anisotropy, where there is no symmetry

1.4 Isotropic-nematic transition: bulk theoretical approaches between perpendicular planes. The material causes a beam of light to be split into two components, each traveling at different velocities, and thus having different re-fractive indices, dimensionless in value. Birefringence is measured as the difference between these refractive indices, and thus in an isotropic material we expect the birefringence to be zero [52]. A birefringence measurement of the liquid crystal 8CB in bulk is given in Fig. 1.3. In the isotropic phase light passes through the liquid crystal at equal velocities, regardless of the direction of travel. The nematic phase is however anisotropic, and non-zero birefringence is measured.

Some experimental techniques, such as Raman scattering and nuclear magnetic resonance, measure S through the anisotropy of individual molecules rather than the bulk anisotropies of the liquid crystal [53]. Although it is impossible to measure the anisotropy of a single molecule by these techniques, the statistical averages (temporal and spatial) of the molecular anisotropy is possible to measure.